Verstappen stopped again after 44 laps, this time for medium tires, and rejoined between Leclerc and Vettel.
Wolff's own future has been the subject of speculation, with sources linking him to a future role at Formula One's owners Liberty Media or the governing FIA.
Bottas eventually finished second, keeping up his team's record of having recorded one-two finishes in every Grand Prix in the 2019 season.
It was another race that featured a pivotal moment at the start.
"Yesterday was a hard day, but I arrived today and I got this great message from this young kid called Harry and he just became my spirit and that was my inspiration", Hamilton told Sky Sports after winning ahead of Bottas to return to the top of the world championship.
As the German rejoined, he ran team-mate Leclerc out of road, costing the Monegasque the third place he had just before claimed as Verstappen backed off to avoid running into the back of Bottas.
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The Ferrari drivers had twice followed orders by then, swapping places as the Italian team tried in vain to come up with a strategy to close the gap. Ferrari then split its strategies, putting Vettel on a two-stop to match the Mercedes and Red Bulls, while Leclerc adopted a promising-looking one-stop. They stopped him again for new medium tires after Lap 40, Vettel returning to the racetrack P6 behind Pierre Gasly, whose position he retook by the end of the lap.
The race was blown wide open in the final stages, when McLaren rookie Lando Norris collided with Racing Point's Lance Stroll as the pair started their 45th lap. But the German locked up and ran out of road, and had to give up the fight on returning to the track, nearly losing out to teammate Charles Leclerc. As a result, he was shuffled back from third to fifth.
And restart the race did, after a lengthy safety vehicle period, the Mercedes-AMG GT bowing out on lap 52. The restart was well managed by Hamilton, who comfortably held the lead ahead of Bottas. Gasly got alongside Leclerc and nearly made a move stick but lost the battle as the pair went through the first two corners.
The Haas drivers almost tangled before Kevin Magnussen moved clear into seventh behind Pierre Gasly's Red Bull, while his team-mate Romain Grosjean dropped back first behind McLaren's Carlos Sainz and Toro Rosso's Daniil Kvyat before battling to hold off Kvyat's team-mate Alexander Albon in the closing laps for the final point. Vettel fell to fourth at 48 points back.
And, in a touching gesture by the sport's world champions, they sent a Mercedes F1 vehicle to Harry's house.
"When that is not there it is definitely a lot less exciting from a competition point of view, racing within a team.it is not how F1 should be, but it is what it is right now and it has been like that in the past, but it is not our fault these guys (Mercedes) are good at their jobs".